Group #21447529 — subgroups

Subgroups — #21447529

Order 7808.

232 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14519799	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519717	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839565	C₂	1	2	yes	maximal, complemented
4	#16839562	C₂	1	2	yes	maximal, complemented
5	C₂×#107×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11758222	—	2	1	no	—
7	#14519799	—	2	2	no	—
8	#14519799	C₄	1	4	yes	complemented
9	C₂×Dic₂₄₄	C₂²	1	1	yes	characteristic
10	C₂×#11758190	—	2	1	no	—
11	C₂×D₄₈₈	C₂²	1	1	yes	characteristic
12	#14519717	—	2	2	no	—
13	C₂×#11758185	C₂²	1	2	yes	—
14	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
16	#107×C₆₁	—	2	1	no	—
17	#107×C₆₁	C₂²	1	2	yes	—
18	C₂²×C₄×C₆₁	—	4	1	no	—
19	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
20	C₂×C₂²⋊D₆₁	—	4	1	no	—
21	#11758222	—	2	4	no	—
22	#11758222	—	4	2	no	—
23	C₂×D₂₄₄	—	2	1	no	—
24	C₂×C₄×D₆₁	—	4	1	no	—
25	C₂×Dic₁₂₂	—	2	1	no	—
26	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
28	C₂×C₈×C₆₁	D₄	1	2	yes	—
29	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₆₁	—	2	2	no	—
31	C₄⋊C₄×C₆₁	—	4	1	no	—
32	Dic₂₄₄	—	2	1	no	—
33	Dic₂₄₄	C₂×C₄	1	2	yes	complemented
34	D₄₈₈	—	2	1	no	—
35	D₄₈₈	C₂×C₄	1	2	yes	complemented
36	#11758190	—	2	4	no	—
37	#11758185	—	2	2	no	—
38	C₂³×C₆₁	D₈	1	1	yes	characteristic
39	C₂²×D₆₁	—	4	1	no	—
40	C₂×C₄×C₆₁	—	4	2	no	—
41	C₂×C₄×C₆₁	—	8	2	no	—
42	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₆₁	D₈	1	1	yes	characteristic
44	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
45	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂²⋊D₆₁	—	4	4	no	—
47	C₂×Dic₆₁	—	4	1	no	—
48	D₂₄₄	—	2	2	no	—
49	D₂₄₄	—	4	1	no	—
50	C₄×D₆₁	—	4	4	no	—
51	Dic₁₂₂	—	2	2	no	—
52	Dic₁₂₂	—	4	1	no	—
53	C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
54	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₆₁	—	2	1	no	—
57	C₂²×C₆₁	—	2	1	no	—
58	C₂²×C₆₁	#102	1	2	yes	—
59	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
60	C₂×D₆₁	—	4	2	no	—
61	C₂×D₆₁	—	8	1	no	—
62	C₂×D₆₁	—	8	1	no	—
63	C₄×C₆₁	—	8	2	no	—
64	C₄×C₆₁	#102	1	2	yes	—
65	C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
67	Dic₆₁	—	4	2	no	—
68	#2139	—	61	1	no	maximal, Sylow
69	C₂×C₆₁	—	2	2	no	—
70	C₂×C₆₁	C₂×#102	1	1	yes	characteristic
71	C₂×C₆₁	#509	1	1	yes	characteristic
72	C₂×C₆₁	#510	1	1	yes	characteristic
73	D₆₁	—	8	2	no	—
74	C₂×#135	—	61	1	no	—
75	C₂×M₃₂	—	61	1	no	—
76	C₂×#107	D₆₁	1	1	yes	characteristic, complemented
77	#358	—	61	2	no	—
78	#357	—	61	2	no	—
79	C₆₁	#2139	1	1	yes	characteristic, complemented, Sylow
80	C₂×#41	—	122	1	no	—
81	#135	—	61	4	no	—
82	#135	—	122	2	no	—
83	C₂×Q₁₆	—	61	1	no	—
84	C₂×SD₁₆	—	122	1	no	—
85	C₂×D₈	—	61	1	no	—
86	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
87	C₂×C₄⋊C₄	—	2	1	no	—
88	M₃₂	—	122	2	no	—
89	C₂×C₁₆	—	61	2	no	—
90	#107	—	2	1	no	—
91	#107	C₂×D₆₁	1	2	yes	—
92	#41	—	122	4	no	—
93	#41	—	244	2	no	—
94	C₂×Q₈	—	122	1	no	—
95	C₂×D₄	—	122	1	no	—
96	C₂×D₄	—	244	1	no	—
97	C₂²×C₄	—	4	1	no	—
98	C₂²×C₄	—	244	1	no	—
99	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
100	Q₁₆	—	61	2	no	—
101	Q₁₆	—	122	1	no	—
102	SD₁₆	—	122	4	no	—
103	D₈	—	61	2	no	—
104	D₈	—	122	1	no	—
105	C₂×C₈	C₄×D₆₁	1	2	yes	—
106	C₂×C₈	D₂₄₄	1	2	yes	—
107	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
108	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	4	1	no	—
111	C₁₆	—	122	2	no	—
112	C₂³	—	244	1	no	—
113	C₂³	#11758199	1	1	yes	characteristic
114	Q₈	—	122	2	no	—
115	Q₈	—	244	1	no	—
116	D₄	—	122	2	no	—
117	D₄	—	244	4	no	—
118	D₄	—	244	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	8	2	no	—
121	C₂×C₄	—	244	1	no	—
122	C₂×C₄	—	244	4	no	—
123	C₂×C₄	#11758198	1	2	yes	—
124	C₂×C₄	#11758199	1	1	yes	characteristic
125	C₂×C₄	#11758201	1	2	yes	—
126	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
127	C₈	#11758198	1	2	yes	—
128	C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
129	C₈	C₂×D₂₄₄	1	1	yes	characteristic
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	488	1	no	—
133	C₂²	—	488	1	no	—
134	C₂²	—	244	2	no	—
135	C₂²	#14519697	1	2	yes	—
136	C₂²	C₂×#11758199	1	1	yes	characteristic
137	C₄	—	8	2	no	—
138	C₄	—	244	2	no	—
139	C₄	#14519697	1	2	yes	—
140	C₄	C₂×#11758198	1	1	yes	characteristic
141	C₄	C₂×#11758201	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	—	488	2	no	—
144	C₂	C₂×#14519697	1	1	yes	characteristic
145	C₂	#16840198	1	1	yes	characteristic
146	C₂	#16840203	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519799

C₂

yes

maximal, characteristic

C₂×#14519717

C₂

yes

maximal, characteristic, complemented

#16839565

C₂

yes

maximal, complemented

#16839562

C₂

yes

maximal, complemented

C₂×#107×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×#11758222

—

#14519799

—

#14519799

C₄

yes

complemented

C₂×Dic₂₄₄

C₂²

yes

characteristic

C₂×#11758190

—

C₂×D₄₈₈

C₂²

yes

characteristic

#14519717

—

C₂×#11758185

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

—

#107×C₆₁

—

#107×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂×C₂²⋊D₆₁

—

#11758222

—

#11758222

—

C₂×D₂₄₄

—

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

Dic₂₄₄

—

Dic₂₄₄

C₂×C₄

yes

complemented

D₄₈₈

—

D₄₈₈

C₂×C₄

yes

complemented

#11758190

—

#11758185

—

C₂³×C₆₁

D₈

yes

characteristic

C₂²×D₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

D₈

yes

characteristic

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

C₂²⋊C₄

yes

—

C₈×C₆₁

C₂²×C₄

yes

characteristic

C₈×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#102

yes

—

C₂²×C₆₁

C₂×D₈

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#102

yes

—

C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

Dic₆₁

—

#2139

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

C₂×#135

—

C₂×M₃₂

—

C₂×#107

D₆₁

yes

characteristic, complemented

#358

—

#357

—

C₆₁

#2139

yes

characteristic, complemented, Sylow

C₂×#41

—

122

—

#135

—

#135

—

122

—

C₂×Q₁₆

—

C₂×SD₁₆

—

122

—

C₂×D₈

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂×C₄⋊C₄

—

M₃₂

—

122

—

C₂×C₁₆

—

#107

—

#107

C₂×D₆₁

yes

—

#41

—

122

—

#41

—

244

—

C₂×Q₈

—

122

—

C₂×D₄

—

122

—

C₂×D₄

—

244

—

C₂²×C₄

—

C₂²×C₄

—

244

—

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

100

Q₁₆

—

101

Q₁₆

—

122

—

102

SD₁₆

—

122

—

103

D₈

—

104

D₈

—

122

—

105

C₂×C₈

C₄×D₆₁

yes

—

106

C₂×C₈

D₂₄₄

yes

—

107

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

108

C₂×C₈

C₂²×D₆₁

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₁₆

—

122

—

112

C₂³

—

244

—

113

C₂³

#11758199

yes

characteristic

114

Q₈

—

122

—

115

Q₈

—

244

—

116

D₄

—

122

—

117

D₄

—

244

—

118

D₄

—

244

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

244

—

122

C₂×C₄

—

244

—

123

C₂×C₄

#11758198

yes

—

124

C₂×C₄

#11758199

yes

characteristic

125

C₂×C₄

#11758201

yes

—

126

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

127

C₈

#11758198

yes

—

128

C₈

C₂×C₄×D₆₁

yes

characteristic

129

C₈

C₂×D₂₄₄

yes

characteristic

130

C₂²

—

131

C₂²

—

132

C₂²

—

488

—

133

C₂²

—

488

—

134

C₂²

—

244

—

135

C₂²

#14519697

yes

—

136

C₂²

C₂×#11758199

yes

characteristic

137

C₄

—

138

C₄

—

244

—

139

C₄

#14519697

yes

—

140

C₄

C₂×#11758198

yes

characteristic

141

C₄

C₂×#11758201

yes

characteristic

142

C₂

—

143

C₂

—

488

—

144

C₂

C₂×#14519697

yes

characteristic

145

C₂

#16840198

yes

characteristic

146

C₂

#16840203

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11758199
commutator	C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₂×#107×C₆₁	C₂
socle	C₂²×C₆₁	C₂×D₈
hypercenter	C₂×#107	D₆₁
hypocenter	C₆₁	#2139
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#107

subgroup

G/H

center

C₂²

C₂×#11758199

commutator

C₈×C₆₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₂×#107×C₆₁

C₂

socle

C₂²×C₆₁

C₂×D₈

D₆₁

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#107

N	H ≅ G/N	type
2. C₂×#14519717	C₂	semidirect
3. #16839565	C₂	semidirect
4. #16839562	C₂	semidirect
5. C₂×#107×C₆₁	C₂	semidirect
8. #14519799	C₄	semidirect
33. Dic₂₄₄	C₂×C₄	semidirect
35. D₄₈₈	C₂×C₄	semidirect
76. C₂×#107	D₆₁	semidirect
79. C₆₁	#2139	semidirect

H ≅ G/N

type

2. C₂×#14519717

C₂

semidirect

3. #16839565

C₂

semidirect

4. #16839562

C₂

semidirect

5. C₂×#107×C₆₁

C₂

semidirect

8. #14519799

C₄

semidirect

33. Dic₂₄₄

C₂×C₄

semidirect

35. D₄₈₈

C₂×C₄

semidirect

76. C₂×#107

D₆₁

semidirect

79. C₆₁

#2139

semidirect

Subgroups — #21447529

Conjugacy classes of subgroups

Complements